Liquid Droplet Dynamics: Variations on a Theme

1
Liquid Droplet Dynamics Variations on a Theme

• Daniel M. Anderson
• Department of Mathematical Sciences
• George Mason University

• Collaborators
• S.H. Davis, Northwestern University
• M.G. Worster, University of Cambridge
• M.G. Forest, University of North Carolina
• R. Superfine, University of North Carolina
• W.W. Schultz, University of Michigan
• J. Siddique, George Mason University
• E. Barreto, George Mason University
• B. Gluckman, George Mason University/Penn. State
  University

Supported by NASA (Microgravity Science), 3M
Corporation and NSF (Applied
Mathematics DMS-0306996)
2
Free-Boundary Problems in Fluid Dynamics

• the location of the free surface is part of the
  solution
• - surface waves in oceans, lakes

wind-driven waves
3
Free-Boundary Problems in Fluid Dynamics

• the location of the free surface is part of the
  solution
• - surface waves in oceans, lakes

canine-driven waves
wind-driven waves
4
Free-Boundary Problems in Fluid Dynamics
Fluids Spreading on Solids

• free surface with moving contact lines LARGE
  SCALE
• floods, lava flows (gravity)

5
Free-Boundary Problems in Fluid Dynamics
The Great Molasses Flood Boston, MA 1919

• From The Boston Globe, May 28, 1996

About 2 million gallons of raw molasses burst
from a storage tank at the corner of Foster and
Commercial streets about noon on January 15,
1919. The black wave of the sticky substance was
so powerful that it knocked buildings off their
foundations and killed 21 people. Newspapers
described the cleanup effort as nightmarish
6
Free-Boundary Problems in Fluid Dynamics
Fluids Spreading on Solids

• free surface with moving contact lines SMALL
  SCALE
• micro-fluidics, nano-fluidics (surface tension)

1mm
7
Outline of Talk

• Isothermal Spreading Droplet (Plain vanilla)
  
• Greenspan, 1978
• Non-Isothermally Spreading Droplet
• Ehrhard Davis, 1991
• Migrating Droplet
• Smith, 1995
• Evaporating Droplet
• Anderson Davis, 1995
• Freezing Droplet
• Anderson, Worster, Davis, Schultz,
  1996, 2000
• Melting Droplet
• Anderson, Forest Superfine, 2001
• Imbibing Droplet, Rigid Porous Substrate
• Hocking Davis, 2000
• Imbibing Droplet, Deformable Porous Substrate
  
• Anderson, 2005
• Vibrating Droplet
• Vukasinovic, Smith, Glezer, James,
  2003, 2004

8
Spreading Droplet Isothermal
9
Anatomy of a Spreading Droplet
10
Spreading Droplet Full Problem

• In the liquid
• - Navier-Stokes Equations
• Free-Surface Conditions
• - Normal and tangential stress balances
• - Mass balance (kinematic condition)
• Conditions at the solid boundary
• - velocity normal to interface is zero
• - slip allowed in tangential velocity
• Contact-line conditions
• - contact (droplet height is zero)
• - condition on contact angle

air
liquid
solid substrate
GOAL Identify a physical regime that corresponds
to experiments and allows isolation of
important physical effects. Reduce
mathematical model accordingly.
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Thin Film Equations Original Form
12
Thin Film Equations Rescaled-Dimensionless Form
13
Thin Film Equations Lubrication Theory Limit
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Isothermal Spreading Droplet Lubrication Theory
Greenspan, 1978 Ehrhard Davis, 1991, 1993
Haley Miksis, 1991

• Slow flow (Re ltlt 1) and slender geometry, zero
  gravity
• Full problem reduces to an evolution equation
  for the interface shape

Capillary number

• symmetry conditions at
• contact line conditions

where
at
Dussan V. 1979 Ehrhard Davis, 1991, 1993
15
Isothermal Droplet Spreading

• large surface tension and

• Analytical formula for
• interface shape and
• contact line position

Droplet Evolution
16
Spreading Droplet Non-Isothermal
17
Anatomy of a Non-isothermally-Spreading Droplet
Ehrhard Davis, 1991
18
Non-Isothermal Spreading Droplet Lubrication
Theory
Ehrhard Davis, 1991, 1993

• Slow flow, slender geometry, zero gravity,
  temperature-dependent surface tension
• Full problem reduces to an evolution equation
  for the interface shape

Marangoni effects (surface tension gradients)
capillarity (surface tension)
unsteady term
Marangoni number
Biot number (interface heat transfer)

• quasi-steady temperature

• contact line conditions

at contact line
19
Non-Isothermal Spreading Droplet Results
Ehrhard Davis, 1991, 1993

• Thermocapillary forces on interface (Marangoni
  effects)
• drive a flow from warmer regions to colder
  regions
• (surface tension decreases with temperature).
• Spreading is enhanced when substrate is cooled.
• Spreading is retarded when substrate is heated.
• Experiments using paraffin oil and silicone oil
  spreading
• on glass confirm these predictions.

20
Migrating Droplet Non-Isothermal
21
Anatomy of a Migrating Droplet
Smith, 1995
22
Migrating Droplet Lubrication Theory
Smith 1995

• Slow flow, slender geometry, zero gravity,
  temp.-dep surface tension
• Imposed temperature variation along solid
  boundary
• Full problem reduces to an evolution equation
  for the interface shape

Marangoni effects (surface tension gradients)
capillarity (surface tension)
unsteady term

• contact line conditions

at left and right contact lines
NOT SYMMETRIC!
23
Migrating Droplet Results
Smith, 1995

• Droplet placed on a non-uniformly heated
  substrate
• migrates towards colder temperature region
  (for
• sufficiently large temperature gradients).
• Steady-state solutions include motionless drops
  and drops
• moving at constant speed (towards cooler
  regions).
• Thermocapillary-driven fluid flow in the drop
  distorts
• the free surface, modifies the apparent
  contact angle
• which in turn modifies contact line speed.

24
Migrating Droplet Results
Smith, 1995
COLD
HOT
video compliments of Marc Smith, 2006
25
Evaporating Droplet
26
Anatomy of an Evaporating Droplet
27
Evaporating Droplet

• (Anderson Davis, 1994 Hocking 1995).

• Lubrication theory leads to an evolution
  equation

Marangoni effects (surface tension gradients)
evaporation (mass loss)
vapor recoil
capillarity (surface tension)
Evaporation number
Marangoni number
Scaled density ratio
Nonequilibrium param.
Slip coefficient
Capillary number
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Evaporating Droplet

• Anderson Davis, 1994 Hocking 1995.

• Lubrication theory leads to an evolution
  equation

boundary conditions
contact line condition
symmetry at
at
at
liquid volume is not constant in time (droplet
vanishes in finite time)
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Evaporating Droplet

• Small capillary number (large surface tension)
  Anderson Davis, 1994.

where
contact line condition
global mass balance
plus initial conditions

• Competition between spreading and evaporation
• EVAPORATION EVENTUALLY WINS!

30
Evaporating Droplet

• strong evaporation,
• weak spreading

• contact line position recedes
• monotonically
• contact angle increases initially
• and remains relatively constant

31
Evaporating Droplet

• weak evaporation,
• strong spreading

• contact line position advances
• initially
• contact angle decreases
• monotonically and has a nearly
• constant intermediate region

32
Evaporating Droplet Results
Anderson Davis, 1995

• Evaporative effects are strongest near the
  contact-line
• region due to largest thermal gradients
  there.
• Effects that increase the contact angle retard
  evaporation
• - thermocapillarity flow directed
  toward the colder
• droplet center
• - vapor recoil nonuniform pressure
  (strongest at contact
• line) tends to contract the
  droplet
• Effects that decrease the contact angle promote
  evaporation
• - contact line spreading

33
Freezing Droplet
34
Freezing Droplet

• This problem is motivated by the need to
  understand crystal growth problems and
  containerless processing systems such as
  Czochralski growth, float-zone processing or
  surface melting.
• The common feature in these systems is the
  presence of a tri-junction where a liquid,
  its solid and a vapor phase meet at which phase
  transformation occurs.
• Simple Model Problem
• WHAT HAPPENS WHEN WE FREEZE A LIQUID
• DROPLET FROM BELOW ON A COLD SUBSTRATE?

35
Experimental Investigation (Water/Ice)
Anderson, Worster Davis (1996)
Initial, motionless, water droplet at room
temperature
36
Experimental Investigation (Water/Ice)
Anderson, Worster Davis (1996)
Initial, motionless, water droplet at room
temperature
Cool bottom boundary (ethylene glycol
anti-freeze pumped through channels in bottom
plate)
Cool bottom boundary (ethylene glycol
anti-freeze pumped through channels in bottom
plate)
?
37
Experimental Investigation (Water/Ice)
Anderson, Worster Davis (1996)
Initial, motionless, water droplet at room
temperature
Cool bottom boundary (ethylene glycol
anti-freeze pumped through channels in bottom
plate)
Cool bottom boundary (ethylene glycol
anti-freeze pumped through channels in bottom
plate)
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Anatomy of a Freezing Droplet
39
Freezing Droplet

• surface tension dominated liquid shape
  Anderson, Worster Davis, 1996.

Mass balance
Capillarity and gravity relate
Assume the solid liquid interface is planar (1D
heat conduction from cold boundary of temperature
isothermal liquid at temperature )
Tri-junction condition (3 models)
40
Freezing Droplet Constant Contact Angle Model

• Contact angle in liquid is constant

Droplet Evolution

• Solidified Shape Cone!

• no inflexion points
• solid shape is independent of growth rate

41
Freezing Droplet Experimental Evidence

• Solidified silicon in crucible of e-beam
  evaporation system (Phil Adams, LSU, 2005)

42
Freezing Droplet Fixed Contact Line Model

• The tri-junction moves tangent to the liquid
  vapor interface the liqiud contact angle is free
  to vary

Solidified Shapes
concave down (zero slope)
concave down (nonzero slope)
concave up (nonzero slope)

• no inflexion points
• water/ice predicted to have zero slope at top
• solid shape is independent of growth rate

43
Freezing Droplet Nonzero Growth Angle Model
Satunkin et al. (1980), Sanz (1986), Sanz et al.
(1987)

• The tri-junction moves at a fixed growth angle
  to the liquid vapor interface (angle
  through vapor phase is )

Solidified Shapes
concave down (nonzero slope)
concave up (nonzero slope)

• no inflexion point
• all materials with nonzero growth angle have
  pointed top
• solid shape is independent of growth rate

44
Freezing Droplet Nonzero Growth Angle
simulation
Experiment ice
45
Freezing Droplet

• A two-dimensional model for the thermal field in
  the solid was obtained by a boundary integral
  method Schultz, Worster Anderson, 2000.

46
Freezing Droplet
Schultz, Worster Anderson, 2000
Results

• both peaks and dimples
• can form at the top of
• the drop (depending
• on the growth angle
• and density ratio)
• inflexion points are also
• possible

47
Melting Droplet
48
Melting Droplet

• Motivated by experiments on polystyrene spheres
  (1mm radius)
• by D. Glick UNC Physics Ph.D. 1998 with
  R. Superfine

Glick Contact Angle Data

• thermal diffusion time
• 10 25 seconds
• data collapse if time is
• scaled with

138C
99C
viscosity (varies by 3 orders of magnitude in
experiment)
surface tension (varies by 10)
ad hoc length scale, increases with temperature
49
Anatomy of a Melting Droplet
50
Melting Droplet Model
Anderson, Forest Superfine, 2001

• initially spherical solid
• no gravity
• surface tension dominates quasi-steady liquid
  vapor interface
• solid-liquid interface assumed planar

Liquid Shape Spherical
Nine Unknown Functions of Time
angles
lengths
volumes and pressure
51
Melting Droplet Model
Anderson, Forest Superfine, 2001
Thermal Problem
1D thermal diffusion, planar solid-liquid
interface
Motion of Solid
Balance of forces equation of motion for solid
Mass Balance
Contact-line Dynamics
Geometry
Provides five relations between lengths, angles
and volumes
Differential-Algebraic System solved by DASSL
code Brenan, Campbell, Petzold,
1995
52
Melting Droplet Dynamics
Melting Droplet (medium )
characteristic contact-line speed
measures competition between spreading
and melting
characteristic melting speed
small dynamics similar to isothermal
spreading
large dynamics deviate from isothermal
spreading
53
Melting Droplet Dynamics

• contact angle relaxes faster in
• spreading/melting configuration
• results do not collapse with
• rescaling of time

• contact line is less mobile in
• spreading/melting configuration
• spreading promotes melting

54
Melting and Freezing Droplet
55
Melting and Freezing Droplet Dynamics
56
Imbibing Droplet Rigid Porous Substrate
57
Anatomy of a Droplet Imbibing into a Rigid Porous
Substrate
Hocking Davis, 1999, 2000
58
Imbibing Droplet Rigid Porous Substrate
Hocking Davis, 1999, 2000

• slender limit (lubrication theory)
• imbibition is one-dimensional liquid
  penetrates vertically
• only no radial capillarity. The porous
  base is assumed to
• be made up of vertical pores.

Evolution equations for liquid shape and
penetration depth
1D capillary suction flow
porosity suction parameter
porous-base modified slip coefficient
59
Imbibing Droplet Rigid Porous Substrate
Hocking Davis, 1999, 2000
central region solution
Contact angle cannot be written as a
single-valued function of the contact line speed
in contrast to regular spreading.
60
Imbibing Droplet Deformable Porous Substrate
61
Imbibing Droplet Deformable Porous Substrate

• Motivation and Applications
• - swelling of paper/print film in inkjet
  printing
• - soil science
• - infiltration
• - medical science (flows in soft tissue)

Modeling Assumptions
- adopt the simplest description of fluid drop
(Hocking Davis, 2000) - assume 1D
imbibition and 1D substrate deformation
(Preziosi et al. 1996, Barry Aldis, 1992,1993)
- porous material is initially dry with uniform
solid fraction - no gravity
62
Anatomy of a Droplet Imbibing into a Deformable
Substrate
63
Imbibing Droplet Deformable Porous Substrate
Equations in wet/deforming porous material
Preziosi et al. 1996
solid fraction
mass conservation for solid and liquid
liquid velocity
solid velocity
modified Darcy Eq.
liquid pressure
stress equilibrium
combine into single PDE for solid fraction
permeability
solid stress
liquid viscosity
related to boundary values of solid fraction
64
Imbibing Droplet Deformable Porous Substrate
Boundary Conditions
Interior similarity solution
Exterior numerical solution
65
Imbibing Droplet Deformable Porous Substrate
Hocking Davis model for liquid droplet
66
Deformable Substrate Sponge Problem
water dropped onto an initially dry and
compressed sponge (photos by E. Barreto and B.
Gluckman)
67
Deformable Substrate 1D Sponge Problem With
Gravity
Capillary-rise of a liquid into a deformable
porous material. -- How does this compare to
the case of a rigid porous material? -- Does
the liquid rise to an equilibrium height? --
How much deformation occurs?
Ask Javed!!
68
Vibrating Droplet
69
Vibrating Droplet Droplet Atomization
James, Vukasinovic, Smith, Glezer (J. Fluid
Mech. 476, 2003) Vukasinovic, Smith, Glezer
(Phys. Fluids, 16, 2004)
videos compliments of Marc Smith, 2006
From JVSG During droplet ejection, the
effective mass of the dropdiaphragm system
decreases and the resonant frequency increases.
If the initial forcing frequency is above the
resonant frequency of the system, droplet
ejection causes the system to move closer to
resonance, which in turn causes more vigorous
vibration and faster droplet ejection. This
ultimately leads to drop bursting.
70
Other Droplet Work

• Isothermal Spreading
• Hocking, 1992 de Gennes, 1985 Dussan V.
  Davis, 1974
• Shikhmurzaev, 1997 Thompson Robbins,
  1989 Koplik
• Banavar, 1995 Bertozzi et al. 1998,
  Barenblatt et al. 1997,
• Jacqmin, 2000
• Evaporating Drops
• Hocking, 1995 Morris, 1997, 2003, 2004
  Ajaev, 2005
• Freezing Drops
• Ajaev Davis, 2003

• Reactive Spreading
• Braun et al., 1995 Warren, Boettinger
  Roosen, 1998
• Motion and Arrest of a Molten Droplet
• Schiaffino Sonin, 1997
• Evaporating and Migrating Droplet
• Huntley Smith, 1996
• Spreading of Hanging Droplets
• Ehrhard, 1994

AND MANY OTHERS!!!
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Summary

• The plain vanilla droplet spreading problem
  and its
• multiple variations lead to interesting
  scientific,
• experimental, mathematical modeling and
  computational
• problems in the general class of free-boundary
  problems
• in fluid mechanics and materials science.

• There are lots of variations still to explore!

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The End

• This work has been supported by
• - National Aeronautics and Space
  Administration (NASA)
• Microgravity Science and Application
  Program
• - 3M Corporation
• - National Science Foundation
• (Applied Mathematics Program,
  DMS-0306996)